Communicating to Learn
Communication as a Learning Tool
Mathematics is a special language consisting of words, tables, graphs and symbols to represent and communicate mathematical ideas. As with all language acquisition, students learn to communicate mathematics by talking, listening, reading, and writing. In CMP classes students build the skills that allow them to communicate their thinking with many others. They make conjectures, try these out, report on progress and refine their thinking.
Aspects of Research from the Cognitive Sciences
An important aspect of communication is possession of a repertoire of representations. The ability to represent mathematical knowledge in a variety of ways is an important indication of students' connected mathematical knowledge. CMP authors have interpreted this theory to imply that curriculum materials should frequently provide, and ask for, knowledge representation using graphs, number patterns, written explanations, and symbolic expressions. These activities facilitate both thinking and communicating.
There is a consistent and substantial body of research indicating that when students engage in cooperative work on appropriate problem solving tasks, their mathematical learning and communication skills will be enhanced. CMP authors have interpreted this line of theory and research to imply that they should design student and teacher materials that are suitable for use in cooperative learning instructional formats, as well as individual learning formats. The particular mathematical task dictates the format. The overarching goal is to make sense of and take ownership of mathematical concepts. This goal is more efficiently reached when students are given opportunities to discuss their thinking with peers and teachers. The mental activity of formulating, representing, clarifying, communicating and reflecting on ideas leads to an increase in learning.
This emphasis on communication is a fairly new development in this country. In some other countries this emphasis on communication is not so new. Watching TIMMS videos (9) of Japanese classrooms or reading about Chinese classrooms in Liping Ma's book (6), the observer is struck by how much careful communicating is going on, far more elaborate than the short answers quickly given that parents and guardians may recall from their own schooldays.
The Parent/Guardian Role
Parents and guardians can also be part of this communication process when they allow students to explain what they have learned and where they still have difficulty. Having your child teach you a concept learned in class is a powerful way of reinforcing that learning, and enhancing achievement and communication skills. Homework questions consciously ask for explanation and reflection, all with the goal of advancing the mathematics under study.
Communication is integral to a CMP classroom, both as a conceptual development tool, and as a means for the teacher to assess what each student knows. Thus an observer would see teachers monitoring group investigations, or leading class discussions, making informal assessments of individuals and the whole group, and adjusting their plans as they gather information.
In a CMP classroom an observer would see students poring over their individual and collaborative work, making suggestions for improvements, and in the process making their own sense of the ideas being studied. "Effective learning environments are community-centered. These communities can build a sense of comfort with questioning rather than knowing answers and can develop a model of creating new ideas that builds on the contributions of individual members." (Pellegrino)
National Research Council. How People Learn: Brain, Mind, Experience, and School. Committee on Developments in the Science of Learning and the Committee on Learning Research and Educational Practice. J Bransford, A. Brown, R. Cocking, S. Donovan, and J. Pellegrino (eds.).Washington, DC: National Academy Press 2000.
Garafolo, Joe and Frank K Lester, Jr. "Metacognition, Cognitive Monitoring, and Mathematical Performance." Journal for Research in Mathematics Education 16 (May 1985): 163-76.
Hiebert, James. "Relationships between Research and the NCTM Standards." Journal for Research in Mathematics Education 30 (January 1999): 3 - 19.
Lampert, Magdalene. "When the Problem is not the Question and the Solution is Not the Answer: Mathematical Knowing and Teaching." American Educational Research Journal 27, no. 1 (Spring 1990): 29-63
Lampert, Magdalene, and Paul Cobb. "Communications and Language." In a Research Companion to NCTM's Standards, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter. Reston Virginia: National Council of teachers of Mathematics, 2003
Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates, 1999.
Silver, Edward A., Jeremy Kilpatrick, and Beth G. Schlesinger. Thinking Through Mathematics: Fostering Enquiry and Communication in Mathematics Classrooms. New York: College Entrance Examination Board, 1990.
Silver, Edward A., and Margaret S. Smith. "mplementing Reform in the Mathematics Classroom: Creating Mathematical Discourse Communities." In Reform in Math and Science Education: Issues for Teachers. Columbus, Ohio: Eisenhower National Clearing House for Mathematics and Science Education, 1997. CD-ROM.
Stigler, James W., and James Heibert The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom. New York: The Free Press, 1999.